What is the value of x in the equation (a/b)^(2x-3) = (b^3/a^3)^(x+4)?

3 Answers
Ashley H
Aug 11, 2018

#x=-9/5#

Explanation:

Provided information is

#(a/b)^(2x-3) = (b^3/a^3)^(x+4)#

so we can rearrange it as,

#(a/b)^(2x-3)=(b/a)^(3(x+4))=(b/a) ^(3x+12)=(a/b)^(-3x-12)#

{ as, #1/b^(-x) =b^0/b^-x=b^(0-(-x))=b^x# similarly, #1/a^x# can be written as #a^0/a^x=a^(0-x)=a^-x#}

Now,comparing,both sides,

#2x-3=-3x-12#

or, #5x=-9#

or, #x=-9/5#

Jumbotron
Aug 11, 2018

#x = -9/5#

Explanation:

#(a/b)^(2x-3) = (b^3/a^3)^(x+4)#

Recall;

#color(indigo)((x^a/y^a) = (x/y)^a)#

Hence;

#(a/b)^(2x-3) = (b/a)^(3(x+4)#

Simplifying;

#(a/b)^(2x-3) = (b/a)^(3x+12)#

Also recall;

#color(indigo)(y/x = 1/(x/y) or (x/y)^-1)#

Hence;

#(a/b)^(2x-3) = 1/(a/b)^(3x+12)#

Simplifying;

#(a/b)^(2x-3) = (a/b)^(-1(3x+12)#

#(a/b)^(2x-3) = (a/b)^(-3x-12)#

#cancel((a/b))^(2x-3) = cancel((a/b))^(-3x-12)#

#2x - 3 = -3x - 12#

Collecting like terms;

#2x + 3x = -12 + 3#

#5x = -9#

#x = -9/5#

P dilip_k
Aug 11, 2018

# (a/b)^(2x-3) = (b^3/a^3)^(x+4)#

# =>a^(2x-3)xxa^(3x+12) = b^(3x+12)xxb^(2x-3)#

# =>a^(5x+9) = b^(5x+9)#

# =>a^(5x+9) / b^(5x+9)=1#

# =>(a/b)^(5x+9) =(a/b)^0#

#=>5x+9=0#

#=>x=-9/5#

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